Skip to main content
SpringerLink
Log in

Derivatives and exceptional poles of the local exterior square L-function for \(GL_m\)

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let \(\pi \) be an irreducible admissible representation of \(GL_m(F)\), where F is a non-archimedean local field of characteristic zero. In 1990’s Jacquet and Shalika established an integral representation for the exterior square L-function. We complete, following the method developed by Cogdell and Piatetski-Shapiro, the computation of the local exterior square L-function \(L(s,\pi ,\wedge ^2)\) via the integral representation in terms of L-functions of supercuspidal representations by a purely local argument. With this result, we show the equality of the local analytic L-functions \(L(s,\pi ,\wedge ^2)\) via the integral representation for the irreducible admissible representation \(\pi \) for \(GL_m(F)\) and the local arithmetic L-functions \(L(s, \wedge ^2(\phi (\pi )))\) of its Langlands parameter \(\phi (\pi )\) through local Langlands correspondence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anandavardhanan, U.K., Kable, A.C., Tandon, R.: Distinguished representations and poles of twisted tensor \(L\)-functions. Proc. Am. Math. Soc. 132(10), 2875–2883 (2004)

    Article  MathSciNet  Google Scholar 

  2. Banks, W.D.: A corollary to Bernstein’s theorem and Whittaker functionals on the metaplectic group. Math. Res. Lett. 5(6), 781–790 (1998)

    Article  MathSciNet  Google Scholar 

  3. Belt, D.D.: On the Holomorphy of Exterior-Square L-Functions, Thesis (Ph.D.), 78, Purdue University (2012). arXiv:1108.2200v7

  4. Bernstein, I.N., Zelevinsky, A.V.: Representations of the group \(GL(n,F),\) where \(F\) is a local non-archimedean field. Russ. Math. Surv. 31(3(189)), 1–68 (1976)

    Article  MathSciNet  Google Scholar 

  5. Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. I. Ann. Sci. École Norm. Sup. (4) 10(4), 441–472 (1977)

    Article  MathSciNet  Google Scholar 

  6. Casselman, W., Shahidi, F.: On irreducibility of standard modules for generic representations. Ann. Sci. École Norm. Sup. (4) 31(4), 561–589 (1998)

    Article  MathSciNet  Google Scholar 

  7. Cogdell, J.W., Matringe, N.: The functional equation of the Jacquet-Shalika integral representation of the local exterior-square \(L\)-function. Math. Res. Lett. 22(3), 697–717 (2015)

    Article  MathSciNet  Google Scholar 

  8. Cogdell, J.W., Piatetski-Shapiro, I.I.: Exterior square \(L\)-function for GL(n). Talk given at the Fields Institute, April (1994). http://www.math.osu.edu/~cogdell.1/

  9. Cogdell, J.W., Piatetski-Shapiro, I.I.: Derivatives and L-Functions for GLn, Representation Theory, Number Theory, and Invariant Theory. In: Progr. Math., vol. 323, Birkhäuser/Springer, Cham, pp. 115–173 (2017)

  10. Field, M.J.: Several complex variables and complex manifolds. I. London Mathematical Society Lecture Note Series, vol. 65. Cambridge University Press, Cambridge-New York (1982)

  11. Gel’fand, I.M., Každan, D.A.: Representations of the Group GL(n,K) Where K is a Local Field, Lie Groups and Their Representations. In: Proc. Summer School, Bolyai Janos Math. Soc., Budapest, pp. 95–118 (1971)

  12. Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. In: Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton (2001) (With an appendix by Vladimir G. Berkovich)

  13. Henniart, G.: Une caractérisation de la correspondance de langlands locale pour \({\rm GL}(n)\). Bull. Soc. Math. France 130(4), 587–602 (2002)

    Article  MathSciNet  Google Scholar 

  14. Henniart, G.: Correspondance de langlands et fonctions \(L\) des carrés extérieur et symétrique. Int. Math. Res. Not. IMRN 4, 633–673 (2010)

    MATH  Google Scholar 

  15. Jacquet, H.P.S., Iosifovitch, I., Shalika, J.: Rankin–Selberg convolutions. Am. J. Math. 105(2), 367–464 (1983)

    Article  MathSciNet  Google Scholar 

  16. Jacquet, H., Piatetski-Shapiro, I.I., Shalika, J.: Automorphic forms on \({\rm GL}(3)\). I and II. Ann. Math. (2) 109(1), 169–258 (1979)

    Article  MathSciNet  Google Scholar 

  17. Jacquet, H., Rallis, S.: Uniqueness of linear periods. Compos. Math. 102(1), 65–123 (1996)

    MathSciNet  MATH  Google Scholar 

  18. Jacquet, H., Shalika, J.: Exterior square L-functions, automorphic forms, shimura varieties, and L-functions, vol. II (Ann Arbor, MI, Perspect. Math., vol. 11. Academic Press, Boston), vol. 1990, pp. 143–226 (1988)

  19. Jo, Y.: The computation of local exterior square L-functions for GLm via Bernstein-Zelevinsky derivatives. Thesis (Ph.D.), The Ohio State University (2018)

  20. Kewat, P.K.: The local exterior square \(L\)-function: holomorphy, non-vanishing and Shalika functionals. J. Algebra 347, 153–172 (2011)

    Article  MathSciNet  Google Scholar 

  21. Kewat, P.K., Raghunathan, R.: On the local and global exterior square \(L\)-functions of \({\rm GL}_n\). Math. Res. Lett. 19(4), 785–804 (2012)

    Article  MathSciNet  Google Scholar 

  22. Kudla, S.S.: The local Langlands correspondence: the non-archimedean case, motives (Seattle, WA, Proc. Sympos. Pure Math., vol. 55) Am. Math. Soc. Provid. 1994, 365–391 (1991)

  23. Łojasiewicz, S.: Introduction to Complex Analytic Geometry. Birkhäuser, Basel (1991) (Translated from the Polish by Maciej Klimek)

    Chapter  Google Scholar 

  24. Matringe, N.: Conjectures about distinction and local Asai \(L\)-functions. Int. Math. Res. Not. IMRN 9, 1699–1741 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Matringe, N.: Distinguished representations and exceptional poles of the Asai-\(L\)-function. Manuscripta Math. 131(3–4), 415–426 (2010)

    Article  MathSciNet  Google Scholar 

  26. Matringe, N.: Distinguished generic representations of \({\rm GL}(n)\) over \(p\)-adic fields. Int. Math. Res. Not. IMRN 1, 74–95 (2011)

    Article  MathSciNet  Google Scholar 

  27. Matringe, N.: Linear and Shalika local periods for the mirabolic group, and some consequences. J. Number Theory 138, 1–19 (2014)

    Article  MathSciNet  Google Scholar 

  28. Matringe, N.: On the local Bump-Friedberg \(L\)-function. J. Reine Angew. Math. 709, 119–170 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Matringe, Nadir: Shalika periods and parabolic induction for \(GL(n)\) over a non-archimedean local field. Bull. Lond. Math. Soc. 49(3), 417–427 (2017)

    Article  MathSciNet  Google Scholar 

  30. Miyauchi, M.: Local newforms and formal exterior square \(L\)-functions. Int. J. Number Theory 9(8), 1995–2010 (2013)

    Article  MathSciNet  Google Scholar 

  31. Rodier, F.: Whittaker models for admissible representations of reductive p-adic split groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), Am. Math. Soc., Providence, R.I., pp. 425–430 (1973)

  32. Shahidi, F.: Fourier transforms of intertwining operators and Plancherel measures for \({\rm GL}(n)\). Am. J. Math. 106(1), 67–111 (1984)

    Article  MathSciNet  Google Scholar 

  33. Shahidi, F.: A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups. Ann. Math. (2) 132(2), 273–330 (1990)

    Article  MathSciNet  Google Scholar 

  34. Shahidi, F.: Twisted endoscopy and reducibility of induced representations for \(p\)-adic groups. Duke Math. J. 66(1), 1–41 (1992)

    Article  MathSciNet  Google Scholar 

  35. Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. II. on irreducible representations of \({\rm GL}(n)\). Ann. Sci. École Norm. Sup. (4) 13(2), 165–210 (1980)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was completed as an extension of the study embarked by Cogdell and Piatetski-Shapiro [8] and owes much to [9] and [28]. This paper is the part of the author’s Ph.D dissertation. He would like to thank his advisor Professor James W. Cogdell for his encouragement and invaluable comments throughout, which helped to improve the exposition of earlier version. Without his help, this paper never have come into existence. Thanks are also due to Professor Nadir Matringe for suggesting me how to prove Proposition 6.1 in [27] and detailed comments to significantly enhance the accuracy of the original manuscript. The author would like to express our sincere appreciation to the referee making incredible suggestions which improved the readability and clarity of this article.

Author information

Author notes

  1. Yeongseong Jo

    Present address: Department of Mathematics, The University of Iowa, Iowa City, IA, 52242, USA

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Yeongseong Jo

Authors
  1. Yeongseong Jo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yeongseong Jo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was partially supported by National Science Foundation (NSF) Grant DMS-0968505 through Professor Cogdell.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jo, Y. Derivatives and exceptional poles of the local exterior square L-function for \(GL_m\). Math. Z. 294, 1687–1725 (2020). https://doi.org/10.1007/s00209-019-02327-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02327-4

Keywords

Mathematics Subject Classification

Search

Navigation